The vertices of quadrilateral PQRS are listed.
P(3,7), Q(6,-2), R(0,-4), S(-3,5)
Which of the following is the strongest classification that identifies quadrilateral PQRS?
Based on the given vertices, we can classify quadrilateral PQRS as a parallelogram.
To prove that quadrilateral PQRS is a parallelogram, we need to show that both pairs of opposite sides are parallel. We can do this by calculating the slopes of the sides.
The slope of the side PQ can be found using the formula:
m = (y2 - y1) / (x2 - x1)
m(PQ) = (-2 - 7) / (6 - 3)
= -9 / 3
= -3
The slope of the side RS can be found using the formula:
m(RS) = (5 - (-4)) / (-3 - 0)
= 9 / -3
= -3
Since the slopes of both sides PQ and RS are equal (-3), they are parallel.
Similarly, we can calculate the slopes of the other pair of sides QR and PS:
m(QR) = (-4 - (-2)) / (0 - 6)
= -2 / -6
= 1/3
m(PS) = (5 - 7) / (-3 - 3)
= -2 / -6
= 1/3
Since the slopes of both sides QR and PS are equal (1/3), they are also parallel.
Therefore, quadrilateral PQRS has both pairs of opposite sides parallel, which makes it a parallelogram.
since the pairs of sides are perpendicular, it is also a rectangle.
Apologies for the mistake in the previous response. You are correct. Based on the given vertices, quadrilateral PQRS can be classified as a rectangle.
To prove that quadrilateral PQRS is a rectangle, we need to show that both pairs of opposite sides are parallel and that the diagonals are congruent and bisect each other.
We have already established that both pairs of opposite sides PQ and RS, as well as QR and PS, are parallel.
To show that the diagonals are congruent, we need to calculate their lengths.
The length of diagonal PR can be found using the distance formula:
d(PR) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((0 - 3)^2 + (-4 - 7)^2)
= sqrt(9 + 121)
= sqrt(130)
The length of diagonal QS can also be found using the distance formula:
d(QS) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((-3 - 6)^2 + (5 - (-2))^2)
= sqrt((-9)^2 + 7^2)
= sqrt(81 + 49)
= sqrt(130)
Since both diagonals PR and QS have the same length (sqrt(130)), they are congruent.
To show that the diagonals bisect each other, we can calculate the midpoints of the diagonals and check if they coincide.
The midpoint of diagonal PR can be calculated as:
M_PR(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)
= ((3 + 0) / 2, (7 + (-4)) / 2)
= (3/2, 3/2)
The midpoint of diagonal QS can be calculated as:
M_QS(x, y) = ((x1 + x2) / 2, (y1 + y2) / 2)
= ((6 + (-3)) / 2, (-2 + 5) / 2)
= (3/2, 3/2)
The midpoints of both diagonals coincide at (3/2, 3/2), indicating that the diagonals bisect each other.
Therefore, quadrilateral PQRS has both pairs of opposite sides parallel, the diagonals are congruent, and the diagonals bisect each other, making it a rectangle.